Geometry And Topology

Mentálne reprezentácie priestoru a ich súvislosť s priestorovým správaním človeka

by Katarína Böhmová

Bachelor thesis

One of the most frequent human activities is targeted moving, that has a condition of using mental capacity assigned... more One of the most frequent human activities is targeted moving, that has a condition of using mental capacity assigned to processing spatial perceptions – using of mental representations of space. In the work there are characteristics of cognitive maps and their connections to the frequency of using elaborated. Emanating from Gallistel, Lynch, Murakoshi & Kawai and Coshall there is hypothesis about more precise topological representation together with high using frequency testified. Second hypothesis of this work – the more frequent ways are represented as shorter – is testified on the part of respondents only. Hypotheses are testified / contradicted on the basis of interviews analyzing, observations, but sketch maps mostly – geometry (lengths, angles) and topology. In the discussion there are analyzed implications resulting from analysis: acquiring knowledge about geometry and topology of space and differences between acquiring them, mental representations relating to woman and mothers on maternity leave as a special category and at last reasoning of inaccurate right angle drawing by students.

Keywords: cognitive maps, mental representations of space, frequency of using ways, geometry, topology, lengths, angles, angles sizes, often, unusual, mothers, students, women, sketch maps, map analysis

Jednou z najčastejších činností človeka je cielený pohyb, ktorého podmienkou je používanie mentálnej kapacity určenej na spracovanie vnemov z priestoru – používanie mentálnych reprezentácií priestoru. Práca rozoberá vlastnosti kognitívnych máp a ich spojitosť s frekvenciou používania ciest. Vychádzajúc z Gallistela, Lyncha, Murakoshiho & Kawaia a Coshalla je v práci na základe analýzy dát terénneho výskumu potvrdená hypotéza o presnejšej topologickej reprezentácii pri vysokej frekvencii používania ciest. Druhá hypotéza tejto práce – častejšie používané cesty reprezentované ako kratšie – je potvrdená iba na časti respondentoch. Hypotézy sú potvrdené / vyvrátené na základe analýzy interview, pozorovaní, ale najmä analýzy náčrtových máp – geometrie (vzdialenosti, uhly) a topológie. V diskusii sú rozobraté implikácie plynúce z výsledkov analýzy: osvojovanie si geometrie a topológie priestoru a rozdiely medzi nimi, mentálne reprezentácie týkajúce sa žien a mamičiek na materskej dovolenke ako osobitnej kategórie a na poslednom mieste zdôvodnenie nepresného zakresľovania pravého uhla študentmi.

Kľúčové slová: kognitívne mapy, mentálne reprezentácie priestoru, frekvencia používania ciest, vzdialenosti, dĺžky, uhly, veľkosti uhlov, častejšie cesty, zriedkavejšie cesty, mamičky, študenti, ženy, náčrtové mapy, analýza máp

DOPO EUCLIDE: ARCHITETTURE E SUGGESTIONI DALLE “ALTRE” GEOMETRIE

by michela rossi

co-authored with Cecilia Tedeschi, presented in the international workshop "Genesi dell'architettura . strumenti per il progetto", Firenze, 29 febbraio-2 marzo 2008.

Geometry, that was born to study and decribe forms, has always been related to architecture. Architecture history... more Geometry, that was born to study and decribe forms, has always been related to architecture. Architecture history shows how it has been as well element of design and measuring, thus an important tool for structure control. In facts knowledge in geometry conditioned the building and, before that, the project; so new cognitions influences architecture and may clear the way to new formal developements. One of the first examples was the use of ellipse, and short later ovals, in anphiteatres plan after Apollonio’s study of conics. Later, geometry studies offered new topics to architects formal research, such as perspective applications in quadrature and anamorphosis in the baroque architecture, or the analityc developments of quadric surfaces designed by contemporary engeneers (Nervi, Candela, Torroja, etc.). Each time the new discoveries acted on architecture both suggesting the developement new forms and offering new project instruments. So mathematicians work leaves its spurs on contemporary architecture, but nineteenth century studies of non Euclidean geometry, due to the not easy representation of n-dimension spaces. Only in last decades of twentieth century the birth of computer graphics allowed the virtual building of objects related to “other” geometry, such as hypercube and topological surfaces; later the computer aided design permitted the planning of architectures with quite innovative formes, such as Gerhy’s and Eisenmann’s ones. But before that we may find spaces related to non Euclidean geometry in M. C. Escher’s graphic work, who developed his lifelong research working together with mathematicians and first offered seeing representation of their concepts.

References
Giuseppa di Cristina, Architettura e Topologia. Per una teoria spaziale dell’architettura, Editrice Librerie Dedalo, 2004.
Nicoletta Sala, Gabriele Cappellato, Architetture della complessità, la geometria frattale tra arte, architettura e territorio, Franco Angeli, Milano, 2004.
Massimiliano Ciammaichella, Architettura in Nurbs, Il disegno digitale della deformazione, Testo & immagine, Torino 2002.
Michele Emmer, La quarta dimensione (euclidea): matematica e arte, in Matematica e Cultura 2001, Sprinter Italia, Milano, 2001, pagg. 201-215
Bruno Ernst, Lo specchio magico di Escher, Taschen Verlag, Berlino, 1990.

Algorithmically detecting the bridge number of hyperbolic knots

by Alexander Coward

We show that, up to ambient isotopy, the exterior of a hyperbolic knot in the 3-sphere admits finitely many bridge... more We show that, up to ambient isotopy, the exterior of a hyperbolic knot in the 3-sphere admits finitely many bridge punctured 2-spheres of given Euler characteristic and that there is an algorithm to find all of these surfaces. This yields an algorithm to detect bridge number for hyperbolic knots.

An upper bound on Reidemeister moves

by Alexander Coward

Submitted

We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the... more We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.

Unknotting genus one knots

by Alexander Coward

Comment. Math. Helv. 86 (2011) 383-399

For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is... more For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and an analysis of the arc complex of the once-punctured torus.

Geometries of Imaginary Space: Architectural Developments of the Ideas of MC Escher and Buckminster Fuller

by michela rossi

Co-authored with Giovanni Ferrero, Celestina Cotti, Cecilia Tedeschi, and presentet to the 6th international Nexus Conference in San Diego (Ca-USA), published on Nexusjournal, 11/2, Birkhauser, Basel, 2009

The long search of M.C. Escher in order to decode the geometric meant of the space among mathematical truth,... more The long search of M.C. Escher in order to decode the geometric meant of the space among mathematical truth, perception and representation, developes itself through some compenetrating topics:
- geometric structure of the landscape, as continuity between morphology and architecture;
- metamorphosis between shapes, ideas or living species in the evolutionary chain;
- rappresentazione of the temporal dimension, also in the ambiguity of the point of view.
The common denominator of this search is the intention to give a shape to the concept of “infinite” that lies toghether unit and variety: the rappresentazione becomes cosmogonia, like a secular image of the creation.
If the sensory knowledge caught up through the perception is illusory, the explanation of the geometric structure of the space through rigorous logic of the mathematical rationality is real and concrete.
The image of the infinite takes shape in the drake biting its tail, in the Moebius’ strips, in the spirals or in the compositions of circles and the squares according to Coxeter’s surface. These forms, sophisticated in their mathematician reference are tied to iniziatory symbols, and ancestral designs.
But the Escher’s graphical inventions are reflected from the work of a contemporary American inventor, who was extremely pragmatic and visionary at the same time. Buckminster Fuller, half architect and half engineer, went on studing spatial simmetries of the platonic solids, in order to resolve the problems of the construction and applies space geometries to the planning of light, industrialized and transportable structures.
To the time of the Cold War its geodetic domes become the symbol of the technology of the West, but his imagination was not pleased of the architectonic construction and he extends this model to the rappresentazione of the land surface, with the licence of one innovative cartographic projection, specifically thought for the air navigation on orthodromic routes.
Fuller’s buildings give concretness to architectural concerns of Escher’s mathematical representations, who trought his graphic constructions based on hiperbolic geometry, opens new ways related to non linear transformations in groups theories. This work means investigate how architectural design may develope new geometries.

References

M.C. Escher, His life and complete graphic work, Abrasale Press, New York, 1982.
Michael J. Gorman, Buckminster Fuller Architettura in movimento, Skira, Ginevra-Milano, 2005
Frucht, R., Graphs of degree three with a given abstract group, Canadian J. Math. 1, pgg. 365-378 (1949)
Klein, F. Vergleichende Betrachtung ueber neue geometriche Forschungen, Program, Erlangen, 1872



References

TQFTs and Higher Dimensional Algebra

by Shay Logan

This paper introduces the tools of higher category theory in a very general and loose way, then demonstrates their use in studying TQFTs.

The curve complex and covers via hyperbolic 3-manifolds

by Robert Tang

C1 fuzzy manifolds

by David Foster

This paper introduces the notion of a C^1 fuzzy manifold as a natural development of the notions of a fuzzy... more This paper introduces the notion of a C^1 fuzzy manifold as a natural development of the notions of a fuzzy topological vector space and of a fuzzy derivative of a fuzzy continuous mapping between fuzzy topological vector spaces. First, a fuzzy atlas of class C^1 on a set is constructed and shown to yield a fuzzy topology that is compatible with the fuzzy atlas. The structure of a C^1 fuzzy manifold on the set then follows. Next, it is shown that the product of two fuzzy manifolds is a fuzzy manifold, and that the composition of two fuzzy differentiable mappings between fuzzy manifolds is fuzzy differentiable. Finally, the notions of a tangent vector and of a tangent space at a point in a fuzzy manifold are formulated, and the tangent space is shown to be a vector space.

Differentiation of fuzzy continuous mappings on fuzzy topological vector spaces

by David Foster

In a classic paper [1] , Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Chang [2], Wong [3],... more In a classic paper [1] , Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Chang [2], Wong [3], Lowen [4], and others developed a theory of fuzzy topological spaces and Rosenfeld [5] initiated a theory of fuzzy groups. These were brought together by Foster [6] to form the elements of a theory of fuzzy topological groups. Starting with a vector space E, a structure for fuzzy vector spaces and fuzzy topological vector spaces was proposed by Katsaras and Liu [7]. In this paper, we develop the theory of fuzzy topological vector spaces further and introduce the notion of the differentiability of fuzzy continuous mappings defined on fuzzy topological vector spaces. The properties of derivatives and formal rules of derivation are also briefly discussed. We point out that our approach does not depend upon the imposition of a norm on the space E. In particular, the derivative defined here should be distinguished from the differential of a "fuzzy function" described by Puri and Ralescu [8l which relates to mappings from an open subset of a normed space into a subset of fuzzy sets defined on a reflexive Banach space.

Fuzzy topological groups

by David Foster

In his classic paper [1] of 1965, Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Subsequently,... more In his classic paper [1] of 1965, Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Subsequently, Chang [2[, Wong [3], Lowen [4] and others applied some basic concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. In an analogous application with groups, Rosenfeld [5] formulated the elements of a theory of fuzzy groups. In the present paper, we bring together the structure of a fuzzy topological space and that of a fuzzy group to form a combined structure, that of a fuzzy topological group. Homomorphic images and inverse images, quotients and products of fuzzy topological groups are also briefly examined. Notation for fuzzy sets follows that of Zadeh [l].

The Classification of Curves

by Jonathan Gleason

The paper I wrote for the Winter 2010 University of Chicago Mathematics DRP.

We introduce the theory of simplicial complexes and use it to prove the triangulability of $1$-manifolds. We then use... more We introduce the theory of simplicial complexes and use it to prove the triangulability of $1$-manifolds. We then use this result to classify all $1$-manifolds.

Ruling Euclidean 3-space

by Evan Siegel